Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ofcc Structured version   Visualization version   GIF version

Theorem ofcc 32745
Description: Left operation by a constant on a mixed operation with a constant. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Hypotheses
Ref Expression
ofcc.1 (𝜑𝐴𝑉)
ofcc.2 (𝜑𝐵𝑊)
ofcc.3 (𝜑𝐶𝑋)
Assertion
Ref Expression
ofcc (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)}))

Proof of Theorem ofcc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ofcc.2 . . . 4 (𝜑𝐵𝑊)
2 fnconstg 6735 . . . 4 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
31, 2syl 17 . . 3 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
4 ofcc.1 . . 3 (𝜑𝐴𝑉)
5 ofcc.3 . . 3 (𝜑𝐶𝑋)
6 fvconst2g 7156 . . . 4 ((𝐵𝑊𝑥𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
71, 6sylan 581 . . 3 ((𝜑𝑥𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
83, 4, 5, 7ofcfval 32737 . 2 (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
9 fconstmpt 5699 . 2 (𝐴 × {(𝐵𝑅𝐶)}) = (𝑥𝐴 ↦ (𝐵𝑅𝐶))
108, 9eqtr4di 2795 1 (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  {csn 4591  cmpt 5193   × cxp 5636   Fn wfn 6496  cfv 6501  (class class class)co 7362  f/c cofc 32734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-ofc 32735
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator